Local limit theorems for conditioned random walks by the heat kernel approximation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912590723022848 |
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| author | Grama, Ion Xiao, Hui |
| author_facet | Grama, Ion Xiao, Hui |
| contents | We study the random walk $(S_n)_{n\geq 1}$ with independent and identically distributed real-valued increments having zero mean and an absolute moment of order $2 + δ$ for some $δ> 0$. For any starting point $x \in \mathbb{R}$, let $τ_x = \inf\{k \geq 1 : x + S_k < 0\}$ denote the first exit time of the random walk $x + S_n$ from the half-line $[0, \infty)$. In the previous work [25], we established a Gaussian heat kernel approximation for both the persistence probability $\mathbb{P}(τ_x > n)$ and the joint distribution $\mathbb{P}(x + S_n \leq \cdot, τ_x > n)$, uniformly over $x \in \mathbb{R}$ as $n \to \infty$. In this paper, we leverage these results to establish a novel conditioned local limit theorem for the walk $(x + S_n)_{n \geq 1}$. For $\mathbb{Z}$-valued random walks, we prove that the joint probability $\mathbb{P}(x + S_n = y, τ_x > n)$ is uniformly approximated by a distribution governed by the Gaussian heat kernel over all $x, y \in \mathbb{Z}$ as $n \to \infty$. Our new asymptotic unifies into a single comprehensive formula the classical local limit theorem by Caravenna [6], as well as various results relying on specific assumptions on $x$ and $y$. As a corollary, we obtain a new uniform-in-$x$ asymptotic formula for the local probability $\mathbb{P}(τ_x = n)$. We also extend our analysis to non-lattice random walks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local limit theorems for conditioned random walks by the heat kernel approximation Grama, Ion Xiao, Hui Probability We study the random walk $(S_n)_{n\geq 1}$ with independent and identically distributed real-valued increments having zero mean and an absolute moment of order $2 + δ$ for some $δ> 0$. For any starting point $x \in \mathbb{R}$, let $τ_x = \inf\{k \geq 1 : x + S_k < 0\}$ denote the first exit time of the random walk $x + S_n$ from the half-line $[0, \infty)$. In the previous work [25], we established a Gaussian heat kernel approximation for both the persistence probability $\mathbb{P}(τ_x > n)$ and the joint distribution $\mathbb{P}(x + S_n \leq \cdot, τ_x > n)$, uniformly over $x \in \mathbb{R}$ as $n \to \infty$. In this paper, we leverage these results to establish a novel conditioned local limit theorem for the walk $(x + S_n)_{n \geq 1}$. For $\mathbb{Z}$-valued random walks, we prove that the joint probability $\mathbb{P}(x + S_n = y, τ_x > n)$ is uniformly approximated by a distribution governed by the Gaussian heat kernel over all $x, y \in \mathbb{Z}$ as $n \to \infty$. Our new asymptotic unifies into a single comprehensive formula the classical local limit theorem by Caravenna [6], as well as various results relying on specific assumptions on $x$ and $y$. As a corollary, we obtain a new uniform-in-$x$ asymptotic formula for the local probability $\mathbb{P}(τ_x = n)$. We also extend our analysis to non-lattice random walks. |
| title | Local limit theorems for conditioned random walks by the heat kernel approximation |
| topic | Probability |
| url | https://arxiv.org/abs/2509.14009 |